It is straightforward to verify that \(\theta \)  is indeed a poset isomorphism.
Therefore \(\Delta (\mathcal {S}^o_{m,k})\)  is isomorphic to
\(\Delta (\mathbf {F}(\mathcal {T}_{m,k}))\) . Note
that \(\Delta (\mathbf {F}(\mathcal {T}_{m,k}))\)  is the (first)
barycentric subdivision of \(\mathcal {T}_{m,k}\)  and
hence it is isomorphic to \(\mathcal {T}_{m,k}\)  (see, e.g. [1]}).
The lemma now follows from the fact that \(\mathcal {T}_{m,k}\) 
is a simplicial sphere with dimension \(k(m-2k-1)-1\) 
(see, e.g. Statement 1.3 and Theorem 1.1 in [2]}).
